Wang, Fan Uniform bound of Sobolev norms of solutions to 3D nonlinear wave equations with null condition. (English) Zbl 1295.35332 J. Differ. Equations 256, No. 12, 4013-4032 (2014). In this paper is considered the time growth rate of the highest Sobolev norm of the solutions to the following nonlinear wave equations \[ u_{tt}-\Delta u=B_{\lambda\mu\nu}\partial_{\lambda}u\partial_{\mu}u\partial_{\nu} u, \tag{1} \]\[ u(0, \cdot)=\phi,\qquad \partial_tu(0, \cdot)=\psi \tag{2} \] in three spatial dimensions, where \(u:[0, T)\times \mathbb{R}^3\longrightarrow \mathbb R\), \(\lambda\), \(\mu\), \(\nu=0, 1, 2, 3\), \(B_{\lambda\mu\nu}\) are constants satisfying the symmetry \(B_{\lambda\mu\nu}=B_{\lambda\nu\mu}\), \((\phi, \psi)\in H_0^s(B_R)\times H_0^{s-1}(B_R)\), \(s\geq 5\), \(B_R\) is a ball of a radius \(R>0\) which center at the origin. The nonlinearities in ({1}) satisfy the null condition \(B_{\lambda\mu\nu}X_{\lambda}X_{\mu}X_{\nu}=0\) for all \(X\in {\mathcal N}\), where \({\mathcal N}=\{X\in \mathbb{R}^4:X_0^2-(X_1^2+X_2^2+X_3^2)=0\}\).The author proves that the problem ({1}), ({2}) satisfying \(E_s(u(0))<\epsilon\) is globally well-posed for a sufficiently small \(\epsilon>0\), and the highest order energy is globally bounded: \(E_s(u(t))\leq C\epsilon\). Reviewer: Svetlin Georgiev (Rousse) Cited in 2 Documents MSC: 35L71 Second-order semilinear hyperbolic equations 35B40 Asymptotic behavior of solutions to PDEs 35L15 Initial value problems for second-order hyperbolic equations Keywords:ghost weight method; time growth rate PDFBibTeX XMLCite \textit{F. Wang}, J. Differ. Equations 256, No. 12, 4013--4032 (2014; Zbl 1295.35332) Full Text: DOI arXiv References: [1] Alinhac, S., The null condition for quasilinear wave equations in two space dimensions I, Invent. 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